The legislation, introduced Tuesday, comes on the heels of a bruising fight at the Board of Supervisors over whether Sheriff Ross Mirkarimi, who was arrested in January on suspicion of domestic violence and later pleaded guilty to one misdemeanor count of false imprisonment, should keep his job; and after a recent a city report finding big spikes in abuse reports between 2010 and 2011.

“The demand for city services and the safety net have increased, while funding is at best flat and in some areas has decreased,” Chiu said, noting the 47 percent increase in calls to domestic violence hotlines and 10 percent surge in cases at the district attorney’s office over the year-long period scrutinized in the report.

Chiu alluded to the Mirkarimi controversy, which culminated in October when four members of the Board of Supervisors rejected Mayor Ed Lee’s attempt to oust him as sheriff, allowing the former supervisor to keep the law enforcement job.

“Obviously over the past year there has been significant public discourse around domestic violence, but despite the recent controversy, everyone believes we need to do more to protect domestic violence victims,” Chiu said Tuesday. “I am trying to move the conversation forward.”

Chiu said he is particularly alarmed by the increased caseload at the District Attorney’s Office, where the number of cases handled per month nearly doubled between 2010 and 2012. Meanwhile, the San Francisco office has less than half the number of dedicated domestic violence staff as Santa Clara County, which handles a similar number of cases.

The proposal, co-sponsored by Supervisors Jane Kim and Malia Cohen, would give the bulk of the money — $445,000 — to the district attorney’s office for four trial attorneys, three investigators, three victim advocates and one paralegal.

District Attorney George Gascón said the extra money “will give us the resources necessary to serve victims,” and “ensure San Francisco remains a city where domestic violence is not tolerated.”

If approved by the board and signed by the mayor, the other $305,000 would go toward civil legal services that help domestic violence survivors do things such as obtain restraining orders and fight child custody battles, to a year-long domestic violence outreach campaign, and to the restoration of a policy analyst position at the Department on the Status of Women.